Learning from Liu Hui? a Different Way to Do Mathematics, Volume 49

Learning from Liu Hui? A Different Way to Do Mathematics Christopher Cullen

ould we have done mathematics differently? But in fact it is not even necessary to imagine At a logical level this question is trivial: re- such an alternative history of mathematics, since Csearch mathematicians spend their time ex- one already exists. Ancient China developed its ploring all the ways one can “do it differently” and own mathematical culture based on a radically dif- then doing them. There are no signs that the math- ferent approach to the structuring of mathemati- ematical enterprise has any artificial barriers round cal thought. Unfortunately, not only western but it that stand in the way of this task. But my ques- also Chinese historians of mathematics have often tion refers to something a little less foundational. failed to see this point. As a result, early Chinese What if we make the “counterfactual move” of try- mathematicians have been portrayed as if they ing to imagine the history of mathematics with were doing the same job as Euclid, only—to be one of its great monuments no longer there—say blunt—a whole lot worse. In history, as in other dis- Euclid, for example. Could we imagine a possible ciplines, using the wrong tools for the job often history of mathematics without Euclid? By “math- breaks the thing you were trying to fix. But what ematics without Euclid” I do not of course mean were ancient Chinese mathematicians up to if they non-Euclidean geometry, but rather a mathematics were not playing the Euclidean game? I hope that stripped of the whole axiomatic-deductive scheme the answer to this question may be of more than for which Euclid’s writing served as the great ex- historical interest, since it bears directly on the emplar and entry point for generations of western pressing question of how mathematicians are to be mathematicians. At first glance the likely course of made and made more effectively. development of such a mathematics seems so dif- The Right Triangle Relation in China ferent from our own that it might deserve a place The problem of what Chinese mathematicians were in one of the more intellectually inclined episodes up to emerges very clearly if we start to ask about of Star Trek (Spock: “It’s mathematics, Jim, but the history of Pythagoras’ theorem in China. not as we know it”). Before we face the main issue, there are a few Christopher Cullen is senior lecturer in the History smaller problems in the way of our shift from west of Chinese Science and Medicine at the School of to east. For a start, we can hardly name the relation Oriental and African Studies, University of London, as used in China after a Greek thinker of around 520 England. His e-mail address is [email protected]. B.C. whose name was not even mentioned in China

AUGUST 2002 NOTICES OF THE AMS 783 until many centuries later.1 More significantly, an- their present form by the first century A.D. In the cient Chinese mathematicians did not talk about Zhou bi the gougu relation is put to practical use right-angled triangles, because they did not talk in four instances,3 while in the Jiu zhang suan shu about triangles in any general sense: there is no an- it is the main theme in all the problems of the cient Chinese term corresponding to Greek 3 trigonon. They did, however, talk quite specifically See sections B11, B29 (twice), and B33 in the version about the ensemble of a horizontal ‘hook’ gou of Cullen (1996). Most of the Zhou bi is concerned with extended out at right angles from the foot of a ver- astronomy rather than mathematics per se. tical line, or ‘leg’ gu . Joining the ends of the hook and leg was what westerners call the hypotenuse, but the Chinese called the ‘bowstring’ xian . One does not, however, usually speak of gouguxian, but just of gougu ‘the hook and the leg’. When was the gougu relation (or whatever we choose to call it) first known and used in China? The first evidence that gou, gu, and xian were known to have a simple and useful relationship is found in the two earliest works of the classical Chinese mathematical canon, the Zhou bi (Gnomon of Zhou), and the Jiu zhang suan shu (Mathematical Methods in a Nine-fold Categorisation).2 Both these works were close to

1It is in any case well known that the “Pythagorean” relation between the sides of a right-angled triangle was used in Mesopotamia long before the study of mathematics began in the Hellenic world. See, for instance, Otto Neugebauer, The Exact Sciences in Antiquity (New York, 1969), pp. 34–6, and plate 6a showing a clay tablet of the Old Babylonian period (c. 1800–1600 B.C.) with the length of the diagonal Figure 1. of a unit square marked with a number equivalent to 1.414213 ... which is in error by one in only the seventh significant figure. As Neugebauer notes, Ptolemy used the same value in computing his table of chords two thousand years later. 2For a full translation and study of the first of these two works, see C. Cullen, Astronomy and Mathe- matics in Ancient China (Cambridge, 1996). The title is often found in the form Zhou bi suan jing (Mathematical canon of the Zhou gnomon), but the words suan jing were not added before approximately A.D. 600. The second work has been translated in Anthony W. C. Lun, J. N. Cross- ley, and Kangshen Shen, The Nine Chapters on the Mathematical Art: Companion and Commentary (Oxford, 1999). The title is translated by Lun et al. in what is now the traditional manner; the version given by me is intended to suggest more clearly the significance of the original Chinese. Figure 2.

784 NOTICES OF THE AMS VOLUME 49, NUMBER 7 ninth and final chapter. A manuscript of a pre- the last sections of this somewhat heterogeneous canonical mathematical text recently found in a work to be written and might have been added to tomb of the second century B.C. does not make use the book to make it more consonant with the cos- of the gougu relation, even where one might have mological numerology popular in the imperial court expected to find it, in problems relating to sawing in the early first century A.D.5 a square beam out of a round log.4 So it seems that However, putting the dating problem to one we have at least a rough fix on when gougu think- side, a further issue arises from the fact that the ing began in China. opening dialogue includes a passage which (though The situation has unfortunately been made much rather obscurely phrased) amounts to no more more confusing by the fact that the book tradi- than the statement of the gougu relation for the tionally placed first in the canonical mathematical case where the ‘hook’ is 3 units, the ‘leg’ is 4 units, series, the Zhou bi, was for centuries commonly and the ‘bowstring’ is 5 units. No premodern Chi- thought to date from the beginning of the Zhou dy- nese commentator has ever claimed to see anything nasty, around 1000 B.C., since it begins with a more substantial here, including some eighteenth- short dialogue between the Duke of Zhou (who and nineteenth-century scholars well versed in ruled as regent near the start of the Zhou dynasty) western as well as Chinese mathematics. Never- and Shang Gao, a sage of the preceding dynasty, theless, a number of twentieth-century historians in which the gougu relation is mentioned. No of mathematics in the East and West have felt scholar now believes in this early dating. I have ar- obliged to extract a “proof of Pythagoras” from this gued that this dialogue is in fact probably one of material by hook or by crook. I think the unfortu- nate results of this will be clear if I exhibit first my 4This text, bearing the title Suan shu shu own fairly literal translation of the relevant part of (A Book of Reckoning and Numbers) was recovered the dialogue, followed by one of the more creative in 1983 from a tomb at Zhangjiashan in versions that have been suggested:6 Hubei province, China. Evidence from other material found there suggests the tomb was closed #A1 [13b] Long ago, the Duke of Zhou in 186 B.C. Like most other Chinese books of its pe- asked Shang Gao , “I have riod, the Suan shu shu was written in ink on a se- heard, sir, that you excel in numbers. ries of bamboo strips bound together with strings, May I ask how Bao Xi 7 laid out the rather like a roller blind. The strings have perished successive degrees of the circumfer- over the centuries, leaving archaeologists with the ence of heaven in ancient times? Heaven tricky task of recovering the original order of 190 cannot be scaled like a staircase, and jumbled strips on which many characters were earth cannot be measured out with a partly or wholly obliterated. These and other diffi- footrule. Where do the numbers come culties no doubt explain why a preliminary tran- from?” scription of this material was published for the first time in 2000. The standard text is now that given #A2 [13f] Shang Gao replied, “The pat- in Peng Hao Zhangjiashan Han jian «Suan shu terns for these numbers come from the shu» zhu shi (The circle and the square. The circle comes Han Dynasty Book on Wooden Strips Suan shu shu from the square, the square comes from found at Zhangjiashan, with a Commentary and Ex- the trysquare, and the trysquare comes planation) (Beijing, 2001). Unlike the highly struc- tured Jiu zhang suan shu, the Suan shu shu lacks 5Cullen (1996), pp. 153–6. any kind of chapter divisions and consists of a se- 6 ries of about 68 problems falling into a dozen or so Translations from the Zhou bi are labelled ac- groupings with broadly common themes. The prob- cording to the system used in Cullen, Astronomy and lems differ in style and terminology to a degree Mathematics in Ancient China (Cambridge, 1996). that suggests they were taken from a variety of 7Bao Xi, also known as Fu Xi , is a mythical fig- sources. The material involving the log is found on ure credited with having conceived the basic struc- pp. 110–3 of Peng’s book. ture of the Yi Jing (Book of Change).

AUGUST 2002 NOTICES OF THE AMS 785 from [the fact that] nine nines are eighty- rectangles like that which has been left one.”8 outside, so as to form a (square) plate. Thus the four (outer) half-rectangles of 9 #A3 [14b] “Therefore fold a trysquare width 3, length 4, and diagonal 5 to- so that the base is three in breadth, the gether make two rectangles (of area 24); altitude is four in extension, and the then (when this is subtracted from the diameter is five aslant. Having squared square plate of area 49) the remainder its outside, halve it [to obtain] one is of area 25. This process is called ‘pil- trysquare.10 Placing them round to- ing up the rectangles’.”12 gether in a ring, one can form three, four, and five. The two trysquares have No comment seems necessary, apart from noting a combined length of twenty-five. This that the insertions and very free interpretations of is called the accumulation of trysquares. the original text evidenced in this version have no Thus we see that what made it possible basis in the Zhou bi, its commentaries, or indeed for Yu 11 to set the realm in order was any other premodern Chinese text. To make mat- what numbers engender.” ters worse, one frequently finds it assumed that the text of the Zhou bi is referring to the diagram re- Here now is a version of the third of these sections, intended to show that there is a proof of produced here as Figure 2, which was in fact only Pythagoras somewhere in there: added—as he tells us himself—by the third-century A.D. commentator Zhao Shuang , who uses it “Thus let us cut a rectangle (diagonally) to illustrate his own essay on the gougu relation, and make the width 3 (units) wide and a piece of writing which is more or less indepen- the length 4 (units) long. The diagonal dent of the main text.13 Zhao Shuang, it may be between the (two) corners will be 5 said, gives a pedestrian explanation of this passage (units) long. Now after drawing a square that in the first place shows that it was no clearer on this diagonal, circumscribe it by half- 12This version, due to Arnold Koslow, is quoted in 8This statement exploits a number of data. The cir- Joseph Needham, Science and Civilisation in China cumference of a unit circle was taken to be three, (Cambridge, 1959), vol. 3, pp. 22–3. I offer this while the perimeter of a unit square is four; “the nine translation as a representative of the lengths to nines” is the name for the traditional multiplication which one may be tempted to go if one is determined table, framed by two rows of numbers at right an- to translate this text as if it was intended to convey gles in the form of the trysquare mentioned below. a proof. I deliberately do not cite other examples (of 9The trysquare ju is the familiar L-shaped car- which there are plenty, some by scholars whose penter’s tool. In an ancient Chinese context it also other work I respect a great deal), and obviously each refers to an L-shaped area made of two rectangu- of them must be judged on its own merits. But I am lar strips at right angles. In the Zhou bi and its com- regretfully convinced that all such efforts amount mentaries it never means a square or rectangle. Al- to making hamburgers without any ground beef to though this shape is sometimes called a “gnomon” put inside the bun. One may admire the ingenuity in western usage, I avoid this term, since I need to of the attempt while declining to eat the result. reserve it for the vertical shadow-casting pole after 13See Cullen (1996), p. 171. What is more, it is clear which the Zhou bi is named. from Zhao’s commentary that the diagram he used 10This is one possible rendering of a sentence which was not in the form seen in most versions of the Zhou is clearly corrupt. None of the variant versions that bi nowadays, in which a 7 by 7 square has four 3- are found in textual sources makes good sense in 4-5 triangles inscribed in its corners, so as to enclose Chinese. See Cullen (1996), p. 94, note 91. an inclined 5 by 5 square in which four further 3- 11A mythical figure whose work as a surveyor and 4-5 triangles are inscribed so as to enclose a unit hydraulic engineer rescued the world from a terri- square. Such a diagram might be used to give a ble flood. graphical dissection proof of the gougu relation,

786 NOTICES OF THE AMS VOLUME 49, NUMBER 7 to him than it is to us and also shows no sign of Guo Shuchun , Shenyang, his seeing a proof of any kind in it.14 1990)16 Up to this point then the record seems to be The rest of the chapter goes on to apply this re- “Greek mathematicians 10, Chinese mathemati- lation to the solution of problems of increasing cians nil”, despite the attempts to smuggle a ball complexity, of which the following is a sample: over the line for the East Asian team. But that is where you get if you mix baseball and football. We need to look more closely at the way Chinese mathematical writing actually functions to see what the Now there is a door whose height is Chinese score actually was. greater than its breadth by 6.8 feet. Two How Chinese Mathematics Worked [diagonally opposite] corners are 10 feet The earliest explicit statement of the relations be- apart. It is asked: what are the height tween the lengths of the gou, gu, and xian occurs and breadth of the door? (Jiu zhang in the Jiu zhang suan shu, an anonymous text suan shu, p. 423) which, as already mentioned, had probably reached The text then gives the solution: the breadth is something close to its present form by the first cen- 2.8 feet, and the height is 9.6 feet.17 It continues: tury A.D.:

Method: Let gou and gu each multiply themselves. Add, and find the side of 15 the square, which is the xian. (Jiu Method: Let the 10 feet multiply itself zhang suan shu, p. 419 in the edition of to make the product. Halve the difference, and let it multiply itself. Double it, subtract from the product. Find the although Zhao does not do this, and neither does any side of the square. With what you ob- other premodern commentator. In fact, it is clear tain, subtract from the halved differ- from his description that his xian tu ‘hy- ence, and that is the breadth of the potenuse diagram’ consisted only of the 5 by 5 door. Add to the halved difference, and square with its inscribed triangles and central unit that is the height of the door. square. The outer part of the usual diagram is never referred to by Zhao and probably originates in the The reader may easily verify algebraically that draftsman’s construction lines used to construct the this method works. No justification of the method inner square and its triangles. is given in the original text, nor is any attempt 14 Zhao’s commentary is fully translated and ex- make use of the same procedure for eliminating that plained in Cullen (1996), pp. 82–8. which corresponds to a given digit in the number 15 “Find the side of the square” is my attempt at a to be divided or whose root is being sought (Professor reasonably faithful but still comprehensible ren- Karine Chemla, REHSEIS Paris, private communi- dering of the conventional Chinese expression Kai cation, May 2002). fang chu zhi , literally “opening the 16Readers may compare my interpretations with square, eliminate it”, which would have been about those in the version of Anthony W. C. Lun, J. N. as puzzling to a nonmathematical ancient Chinese Crossley, and Kangshen Shen, The Nine Chapters reader as our own “find the square root” still is to on the Mathematical Art: Companion and Com- a nonmathematical modern English reader. The expression chu zhi (eliminate it) is probably a ref- mentary, (Oxford, 1999). erence to the parallel between the algorithms for 17For simplicity I have reduced all the units used in square root extraction and division, which both the original text to the chi , roughly one foot.

AUGUST 2002 NOTICES OF THE AMS 787 made to justify the original statement of the gen- http://www.staff.hum.ku.dk/dbwagner/ eral gougu relation. So far the reader may have the Pythagoras/Pythagoras.html feeling that ancient Chinese mathematics consists For our present purposes, the point is that Liu of no more than a body of rules of thumb adopted Hui simply gets this explanation out of the way as blindly and without interest in whether or why the a preliminary to the main business of the chapter: rules worked. Actually the truth is rather stranger this is not a major issue on which he feels the need than that. to dwell.18 His explanation does not use any word Like many ancient Chinese texts, the Jiu zhang that could be mapped onto modern English “proof” suan shu has a commentary, in this instance by the or “theorem” or onto their Greek equivalents. As great mathematician Liu Hui , who was active explicit issues, these were not his concern. What around 260 A.D. What his commentary does is to then was Liu Hui concerned with, as an ancient Chi- follow through all the problems of the book, in ef- nese mathematician? Fortunately he tells us. In his fect taking the “Method” statements of the origi- preface to the commented edition of the Jiu zhang nal text to pieces and reconstructing them in a suan shu, he says: way that enables the reader to see clearly how they work. In the case of the chapter on gougu problems, When I was young I learned the Jiu he does this mainly with reference to diagrams Zhang and when I grew up I went over which are now lost but which can in most cases be it again carefully. I looked into the easily reconstructed. But one case in which the re- breaking apart of Yin and Yang, took a construction is by no means easy is that of the orig- comprehensive view of the basis of inal statement of the gougu relation. Liu Hui’s ex- mathematical methods, and of the sup- planation reads: positions involved in seeking the unknown, and thus attained to realisation of [the work’s] meaning. Therefore I have ventured to exert my meagre ca- pacities to the utmost, and to select from what I have seen [in other books?] The gou multiplied by itself makes the in order to make a commentary. The cat- red square, and the gu multiplied by it- egories under which the matters [treated self makes the blue square. Let there be herein fall] extend each other [when taking away, and putting in, and being compared], so that each benefits [from made complete, each following its kind. the comparison]. So even though the Thus one reaches [a state where] the branches are separate they come from differences no longer are to be adjusted. the same root, and one may know that Together they form the area of the xian they each show a separate tip [of the square. Find the side of this square, and same tree] ( this is the xian. (Jiu zhang suan shu, p. 419) ). (Jiu zhang suan shu, preface, It is clear that Liu Hui assumes his readers can p. 177) see a diagram in which the two smaller squares on And indeed his words seem to echo those found the gou and the gu are in some way dissected and in the second section of the Zhou bi itself, in which reassembled to form the larger square. For him no Chen Zi (a figure unknown to history) is rep- further explanation is necessary. The reader may resented as explaining to his confused student enjoy pausing to try to make such a dissection before turning to one possible and ingenious solution 18He certainly shows no signs of being inclined to suggested by Don Wagner. It was originally pub- sacrifice an ox in celebration, as the legendary west- lished as “A proof of the Pythagorean Theorem by ern account of the theorem’s discovery claims Liu Hui (third century AD)” (Historia Math. 12 Pythagoras did: see Ivor Thomas, Greek Mathe- (1985), pp. 71–3), but may be more conveniently ac- matical Works: I Thales to Euclid (Harvard, 1980), cessed through Wagner’s website at: p. 185, citing Proclus.

788 NOTICES OF THE AMS VOLUME 49, NUMBER 7 Rong Fang the essentials of mathematical sible problems and aimed to show that those known thinking: to him could all be reduced to nine basic categories solvable by nine basic methods. To a great #B6 [24k] Chen Zi replied, “You thought extent he succeeded, although the contents of some about it, but not to [the point of] ma- sections still show a degree of diversity. It did not turity. This means you have not been strike him as worthwhile to try to argue explicitly able to grasp the method of surveying that his methods would always work for the ap- distances and rising to the heights, and propriate problem type. In the first place, he already so in mathematics you are unable to knew they did work—the examples are before us extend categories (tong lei ).… If to this day. Secondly, if it ever turned out that the one asks about one category, and ap- method failed on a new problem, that would not plies [this knowledge] to a myriad af- have been taken as a sign that the method was fairs, one is said to know the Way. wrong, but rather that it was necessary to distin- …Therefore one studies similar meth- guish a new problem category with a new com- ods in comparison with each other, and mon method for all problems of the new type—lei one examines similar affairs in com- yi he lei distinguish categories in order parison with each other. This is what to unite categories, in fact, as Chen Zi says. makes the difference between stupid As a person whose initial mathematical training and intelligent scholars, between the beyond the level of arithmetic was based firmly on worthy and the unworthy. Therefore, it an initiation into a Euclidean structure of axiomatic is the ability to distinguish categories in deduction and stacking one theorem on another,19 order to unite categories (neng lei yi he with problems serving only to show that one had lei ) which is the substance understood the theorems, it is clear to me that the of how the worthy one’s scholarly pat- problem-centred approach of the ancient Chinese rimony is pure, and of how he applies mathematician deals directly with the difficulty himself to the practice of understand- pointed out by one very perceptive student of the ing.” history of science in the West. Students, Thomas What we have here is a concise statement of a Kuhn tells us: twofold heuristic strategy, summed up in the words …regularly report that they have read “distinguish categories in order to unite categories” through a chapter of their text, under- (lei yi he lei ). On the one hand, the math- stood it perfectly, but nonetheless had ematician performs the analytic task of distin- difficulty solving a number of the prob- guishing different problem types, each with their lems at the chapter’s end. Ordinarily, own methods shu from each other. On the , also, those difficulties dissolve in the other hand, the very act of analysis brings together same way. The student discovers, with groups of similar problems which may be treated or without the assistance of his in- synthetically. Further, one can then attempt to structor, a way to see his problem as like “unite categories” at a higher level by finding com- a problem he has already encountered. mon structures underlying different problem cat- Having seen the resemblance, grasped egories. the analogy between two or more dis- Chen Zi’s analytic/synthetic approach is in fact tinct problems, he can interrelate sym- not particularly well exemplified in the Zhou bi it- bols and attach them to nature in the self. It is, however, clearly (or so it seems to me) ways that have proved effective before. the main rationale of the Jiu zhang mentioned ear- … The resultant ability to see a variety lier. Whereas Euclid was concerned to show how a of situations as like each other … is, I great number of true propositions could be de- duced from a small number of axioms, the anony- 19I enjoyed it, by the way; this was Britain in the mous author of the Jiu zhang followed a different late 1950s. Thank you, Mr. B. G. Worsdall (Bedford but no less rational route in the reverse direction. School), for introducing me to the delights of Eu- He started from the almost infinite variety of pos- clidean proof!

AUGUST 2002 NOTICES OF THE AMS 789 think, the main thing a student acquires About the Cover by doing exemplary problems, whether This month's cover accompanies the article by with a pencil and paper or in a well- Christopher Cullen and shows the so-called designed laboratory. After he has com- “hypotenuse diagram” from the Zhou bi, pre- pleted a certain number, which may sumably drawn in its original form by the third- vary widely from one individual to the century commentator Zhao Shuang. It is the next, he views the situations that con- source of the logo of the 2002 International Congress of Mathematicians (ICM), to be held in front him as a scientist in the same Beijing in August 2002. The particular image is gestalt as other members of his spe- a photograph of a page from a manuscript of cialists’ group. For him they are no the late eighteenth century, now located in the longer the same situations he had en- Asian Studies Library of the University of British countered when his training began. He Columbia. has meanwhile assimilated a time-tested This handsome copy was made by hand, pre- sumably from a printed book. The Zhou bi was, and group-licensed way of seeing. as far as we know, the first mathematics book (Thomas Kuhn, The Structure of Scien- ever to be printed. The first edition was pro- tific Revolutions (second edition) duced in the eleventh century, but the earliest (Chicago, 1970), p. 189) extant copy is from the next century, preserved in a library in Shanghai. Printing in China at this Things may be very different in mathematics ed- time was from inked wood blocks, on which ucation today from the situation I recall, in which, the content of a pair of successive pages had as Kuhn hints, the art of problem solving was some- been cut in relief. This technique, which was in- thing one all too often had to discover without the vented in China, played only a small role in Eu- ropean printing, but it was so suitable for the help of one’s teacher. Could it be that students of Chinese writing system, which has a huge char- mathematics in ancient China had a more effective acter set, that it persisted long beyond when introduction to this aspect of their craft than at least movable type had been first introduced. some in the West have received? Since it seems that The coloring scheme is not original, but is well-trained and highly motivated students of math- suggested by the text in the diagram, as shown in Figure 2 of Cullen’s article. The color “red” ematics (let alone teachers of mathematics) are is actually vermilion, an orange-red that since not overly common nowadays, it may still be worth- ancient times has been the color of seal stamps while to ask what there is in the ancient Chinese in much of eastern Asia. approach that might still be useful to us today. Good references are the short book Printing and Publishing in Medieval China by Denis Twitchett and the more comprehensive Paper and Printing by Tsien Tsuen-Hsuin, volume V, part 1, of Science and Civilization in China. —Bill Casselman ([email protected])

790 NOTICES OF THE AMS VOLUME 49, NUMBER 7 AUGUST 2002 NOTICES OF THE AMS 791